Bridging Semisymmetric and Half-Arc-Transitive Actions on Graphs

ISSN： 0195-6698

Source： European Journal of Combinatorics, Vol.23, Iss.6, 2002-08, pp. : 719-732

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

A generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edge- but not vertex-transitive, was given by Marušič and Potočnik (2001, Europ. J. Combinatorics,22, 333–349) together with a natural connection between graphs admitting 1 / 2-arc-transitive group actions and certain graphs admitting semisymmetric group actions. This connection is studied in more detail in this paper. Among others, a sufficient condition for the semisymmetry of the so-called generalized Folkman graphs arising from certain graphs admitting a 1 / 2-arc-transitive group action is given. Furthermore, the concepts of alter-sequence and alter-exponent is introduced and studied in great detail and then used to study the interplay of three classes of graphs: cubic graphs admitting a one-regular group action, the corresponding line graphs which admit a 1 / 2-arc-transitive action of the same group and the associated generalized Folkman graphs. At the end an open problem is posed, suggesting an in-depth analysis of the structure of tetravalent 1 / 2-arc-transitive graphs with alter-exponent 2.